Skip to main content
SpringerLink
Log in

An Asymptotic Mean Value Formula for Eigenvalues of the Hessian Related to Concave/Convex Envelopes

  • Original Article
  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper we characterize viscosity solutions to the PDE λj(D2u) = 0 by means of the asymptotic mean value formula

$$ u(x) = \underset{\dim(S)=j}{\min} \underset{v\in S, |v|=1}{\max} \left\{\frac{1}{2} u (x + \epsilon v) + \frac{1}{2} u (x - \epsilon v)\right\} + o(\epsilon^{2}), $$

as 𝜖 → 0, that holds in the viscosity sense. Here, λ1(D2u) ≤⋯ ≤ λN(D2u) are the ordered eigenvalues of the Hessian D2u. This equation is related to optimal concave/convex envelopes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alexandroff, A.D.: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Leningr. State Univ. Ann. Math. Ser. 6, 3–35 (1939)

    MathSciNet  Google Scholar 

  2. Birindelli, I., Galise, G., Leoni, F.: Liouville theorems for a family of very degenerate elliptic non lineal operators. Nonlinear Anal. 161, 198–211 (2017)

    Article  MathSciNet  Google Scholar 

  3. Birindelli, I., Galise, G., Ishii, I.: A family of degenerate elliptic operators: maximum principle and its consequences. Ann. Inst. H. Poincaré Anal. Non Linéaire 35, 417–441 (2018)

    Article  MathSciNet  Google Scholar 

  4. Blanc, P., Rossi, J.D.: Games for eigenvalues of the Hessian and concave/convex envelopes. J. Math. Pures Appl. 127, 192–215 (2019)

    Article  MathSciNet  Google Scholar 

  5. Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations, III : functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

    Article  MathSciNet  Google Scholar 

  6. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  7. Harvey, F.R., Lawson, H.B. Jr: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62, 396–443 (2009)

    Article  MathSciNet  Google Scholar 

  8. Harvey, F.R., Lawson, H.B. Jr: p-convexity, p-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62, 149–169 (2013)

    Article  MathSciNet  Google Scholar 

  9. Lindqvist, P., Manfredi, J.: On the mean value property for the p-Laplace equation in the plane. Proc. Amer. Math. Soc. 144, 143–149 (2016)

    Article  MathSciNet  Google Scholar 

  10. Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for p-harmonic functions. Proc. Amer. Math. Soc. 138, 881–889 (2010)

    Article  MathSciNet  Google Scholar 

  11. Oberman, A.M., Silvestre, L.: The Dirichlet problem for the convex envelope. Trans. Amer. Math. Soc. 363, 5871–5886 (2011)

    Article  MathSciNet  Google Scholar 

  12. Sha, J.-P.: Handlebodies and p-convexity. J. Differ. Geom. 25, 353–361 (1987)

    Article  MathSciNet  Google Scholar 

  13. Wu, H.: Manifolds of partially positive curvature. Indiana Univ. Math. J. 36, 525–548 (1987)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), by UBACyT grant 20020160100155BA (Argentina) and by MINECO MTM2015-70227-P (Spain).

Author information

Authors and Affiliations

  1. Departamento de Matemática, FCEyN, Universidad de Buenos Aires, and IMAS - CONICET, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina

    Pablo Blanc & Julio D. Rossi

Authors
  1. Pablo Blanc
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julio D. Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julio D. Rossi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

To Marco Antonio López on the occasion of his 70th birthday, with our best wishes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blanc, P., Rossi, J.D. An Asymptotic Mean Value Formula for Eigenvalues of the Hessian Related to Concave/Convex Envelopes. Vietnam J. Math. 48, 335–344 (2020). https://doi.org/10.1007/s10013-020-00385-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-020-00385-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation